Parameter drift in equation error identification

We find situations in which the equation error identifier generates unbounded parameter estimates when driven by bounded sequence. This unexpected parameter divergence, or drift, is related to the inadequacy of excitation of the input sequence and is characterized by slow (i.e. nonexponential) escape of the parameter estimate vector to infinity in spite of all other signals (inputs, outputs, prediction errors) remaining bounded. The analysis proceeds by showing that, in a general adaptive identifier, the sequence of regressors (information vectors) provides a natural decomposition of the parameter estimate space into subspaces, each corresponding to a class of input excitation characteristic of drift behavior. This subspace decomposition is applied to the equation error adaptive identifier, yielding direct links between parameter estimate behavior and modes of excitation. We build on the functional analytic approach of [1] and [14] in which parameter drift in the LMS adaptive filter was discovered. Although the problems encountered in the equation error identifier are related, there are structural differences that distinguish the present analysis the presence of disturbances in the information vector, and the mutual coupling of the components of the information vector through the plant. The "excitation diagram" is introduced as a geometric interpretation for the loss of excitation in certain subspaces Instability (or parameter drift) is proven in certain subspaces of "decaying" excitation. This drift mechanism is inherent in the algorithm and is not due to numerical implementation problems or violation of small step size conditions.